The familiar Euclidean spacesKn, where the Euclidean norm of x = (x1, ..., xn) is given by ||x|| = (∑ |xi|2)1/2, is a Banach space.

The space of all continuous functions f : [a, b]
->K defined on a closed interval [a, b]
becomes a Banach space if we define the norm of such a function as
||f|| = sup { |f(x)| : x in [a, b] }. This is
indeed a norm since continuous functions defined on a closed interval
are bounded. The space is complete under this norm, and the resulting
Banach space is denoted by C[a, b]. This example can be
generalized to the space C(X) of all continuous functions X->K, where X is a compact space, or to the
space of all bounded continuous functions X->K, where X is any topological space, or indeed to the
space B(X) of all bounded functions X->K,
where X is any set. In all these examples, we can multiply
functions and stay in the same space: all these examples are in fact
unitary Banach algebras.

If p ≥ 1 is a real number, we can consider the space of all infinite sequences (x1, x2, x3, ...) of elements in K such that the infinite series ∑ |xi|p converges. The p-th root of this series' value is then defined to be the p-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by l p.

The Banach space l∞ consists of all bounded sequences of elements in K; the norm of such a sequence is defined to be the supremum of the absolute values of the sequence's members.

Again, if p ≥ 1 is a real number, we can consider all functions f : [a, b] ->K such that |f|p is Lebesgue integrable. The p-th root of this integral is then defined to be the norm of f. By itself, this space is not a Banach space because there are non-zero functions whose norm is zero. We define an equivalence relation as follows: f and g are equivalent if and only if the norm of f - g is zero. The set of equivalence classes then forms a Banach space; it is denoted by L p[a, b]. It is crucial to use the Lebesgue integral and not the Riemann integral here, because the Riemann integral would not yield a complete space. These examples can be generalized; see Lp spaces for details.

Finally, every Hilbert space is a Banach space.
The converse is not true.

It is possible to define the derivative of a function f : V->W between two Banach spaces. Intuitively, if x is an element of V, the derivative of f at the point x is a continuous linear map which approximates f near x.

Formally, f is called differentiable at x if there exists a continuous linear map A : V->W such that

limh->0 ||f(x + h) - f(x) - A(h)|| / ||h|| = 0

The limit here is taken over all sequences of non-zero elements in V which converge to 0.
If the limit exists, we write Df(x) = A and call it the derivative of f at x.

This notion of derivative is in fact a generalization of the ordinary derivative of functions R->R, since the linear maps from R to R are just multiplications with real numbers.

If f is differentiable at every point x of V, then Df : V-> L(V, W) is another map between Banach spaces (in general not a linear map!), and can possibly be differentiated again, thus defining the higher derivatives of f. The n-th derivative at a point x can then be viewed as a multilinear map Vn->W.

Differentiation is a linear operation in the following sense: if f and g are two maps V-W which are differentiable at x, and r and s are scalars from K, then rf + sg is differentiable at x with D(rf + sg)(x) = rD(f)(x) + sD(g)(x).

The chain rule is also valid in this context: if f : V->W is differentiable at x in V, and g : W->X is differentiable in f(x), then the composition g o f is differentiable in x and the derivative is the composition of the derivatives:

for all x in V and f in V'. As a consequence of the Hahn-Banach theorem, this map is injective; if it is also surjective, then the Banach space V is called reflexive. Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

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